MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  exists1 Structured version   Visualization version   Unicode version

Theorem exists1 2410
Description: Two ways to express "only one thing exists." The left-hand side requires only one variable to express this. Both sides are false in set theory; see theorem dtru 4592. (Contributed by NM, 5-Apr-2004.)
Assertion
Ref Expression
exists1  |-  ( E! x  x  =  x  <->  A. x  x  =  y )
Distinct variable group:    x, y

Proof of Theorem exists1
StepHypRef Expression
1 df-eu 2323 . 2  |-  ( E! x  x  =  x  <->  E. y A. x ( x  =  x  <->  x  =  y ) )
2 equid 1863 . . . . . 6  |-  x  =  x
32tbt 351 . . . . 5  |-  ( x  =  y  <->  ( x  =  y  <->  x  =  x
) )
4 bicom 205 . . . . 5  |-  ( ( x  =  y  <->  x  =  x )  <->  ( x  =  x  <->  x  =  y
) )
53, 4bitri 257 . . . 4  |-  ( x  =  y  <->  ( x  =  x  <->  x  =  y
) )
65albii 1699 . . 3  |-  ( A. x  x  =  y  <->  A. x ( x  =  x  <->  x  =  y
) )
76exbii 1726 . 2  |-  ( E. y A. x  x  =  y  <->  E. y A. x ( x  =  x  <->  x  =  y
) )
8 nfae 2165 . . 3  |-  F/ y A. x  x  =  y
9819.9 1990 . 2  |-  ( E. y A. x  x  =  y  <->  A. x  x  =  y )
101, 7, 93bitr2i 281 1  |-  ( E! x  x  =  x  <->  A. x  x  =  y )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189   A.wal 1450   E.wex 1671   E!weu 2319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676  df-eu 2323
This theorem is referenced by:  exists2  2411
  Copyright terms: Public domain W3C validator